Microscopic study of low-lying spectra of $\Lambda$ hypernuclei based on a beyond-mean-field approach with covariant energy density functional

We present a detailed formalism of the microscopic particle-rotor model for hypernuclear low-lying states based on a covariant density functional theory. In this method, the hypernuclear states are constructed by coupling a hyperon to low-lying states of the core nucleus, which are described by the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this method to study in detail the low-lying spectrum of $^{13}_{~\Lambda}$C and $^{21}_{~\Lambda}$Ne hypernuclei. We also briefly discuss the structure of $^{155}_{~~\Lambda}$Sm as an example of heavy deformed hypernuclei. It is shown that the low-lying excitation spectrum with positive parity states of the hypernuclei, which are dominated by $\Lambda$ hyperon in $s$-orbital coupled to the core states, are similar to that for the corresponding core states, while the electric quadrupole transition strength, $B(E2)$, from the 2$^+_1$ state to the ground state is reduced according to the mass number of the hypernuclei. Our study indicates that the energy splitting between the first 1/2$^-$ and 3/2$^-$ hypernuclear states is generally small for all the hypernuclei which we study. However, their configurations depend much on the properties of a core nucleus, in particular on the sign of deformation parameter. That is, the first $1/2^-$ and $3/2^-$ states in $^{13}_{~\Lambda}$C are dominated by a single configuration with $\Lambda$ particle in the $p$-wave orbits and thus providing good candidates for a study of the $\Lambda$ spin-orbit splitting. On the other hand, those states in the other hypernuclei exhibit a large configuration mixing and thus their energy difference cannot be interpreted as the spin-orbit splitting for the $p$-orbits.